Excentre

Q. Find the equation of the line whose perpendicular distance from the origin is $5$ units and the angle made by the perpendicular with the positive $x$-axis is ${30}^{\circ }$.

Q. Find the equation of the line whose perpendicular distance from the origin is $4\text{units}$ and the angle which the normal makes with positive direction of $x-$axis is ${15}^{\circ }$.

Q.

Points $\left(\frac{1}{2},\frac{-13}{4}\right)$ divides the line joining the points $\left(3,-5\right)$ and $\left(-7,2\right)$ in what ratio

1. $1:3$ internally

2. $3:1$internally

3. $1:3$ externally

4. $3:1$externally

Q. The vertices of a triangle are $A(0,-6),$ $B(-6,0)$ and $C(1,1)$ respectively. Then coordinates of the excentre opposite to vertex $A$ are

1. $(-\frac{3}{2},-\frac{3}{2})$
2. $(-4,6)$
3. $(-4,\frac{3}{2})$
4. $(\frac{3}{2},-\frac{3}{2})$

Q. Distance of the point $(x_l,y_l,z_l)$ from the line $\frac{x-x_2}{l}=\frac{y-y_2}{m}=\frac{z-z_2}{n}$, where I, m and n are the direction cosines of line is

1. 
   

2. 
   

3. None of these

4. 
   

Q. If $P(x,y)$ divides the line segment joining $A(5,0)$ and $B(10\cos \theta ,10\sin \theta )$ internally in the ratio $2:3,$ then which of the following is (are) true?

1. Distance of $P$ from $(3,0)$ is always constant.
2. $x\in [0,3]$
3. $y\in [-4,0]$
4. $x^2+y^2-6x-7=0$

Q. The vertices of a triangle are $A(0,-6),$ $B(-6,0)$ and $C(1,1)$ respectively. Then coordinates of the excentre opposite to vertex $A$ are

1. $(-\frac{3}{2},-\frac{3}{2})$
2. $(-4,\frac{3}{2})$
3. $(\frac{3}{2},-\frac{3}{2})$
4. $(-4,6)$

Q. If $A(-5,3),B(-3,3)$ and $C(-4,4)$ represents vertices of a triangle, then which of the following option(s) are correct?

1. Incentre of $△ABC$ is $(-4,2+\sqrt{2})$
2. Orthocentre centre of $△ABC$ is $(-4,4)$
3. Excentre opposite to vertex $A$ is $(\sqrt{2}-4,4)$
4. Excentre opposite to vertex $C$ is $(-4,2-\sqrt{2})$

Q. If the vertices of a triangle are $A(2,4),B(2,6)$ and $C(2+\sqrt{3},5)$, then which of the following is/are true?

1. Coordinates of the excentre opposite to vertex $A$ is $(2+\sqrt{3},7)$
2. Coordinates of the excentre opposite to vertex $B$ is $(2-\sqrt{3},3)$
3. Coordinates of the excentre opposite to vertex $C$ is $(2-\sqrt{3},5)$
4. The area of $△ABC$ is $\sqrt{3}\text{sq. units}$

Q. A straight line $L$ through the point $(3,–2)$ is inclined at an angle ${60}^{\circ }$ to the line $\sqrt{3}x+y=1$ If $L$ also intersects the x-axis, then the equation of $L$ is

1. $\sqrt{3}y-x+3+2\sqrt{3}=0$
2. $y+\sqrt{3}x+2-3\sqrt{3}=0$
3. $\sqrt{}y+x-3+2\sqrt{}=0$
4. $y-\sqrt{3}x+2+3\sqrt{3}=0$

Q. If $\lambda$ is the number of $7$ letter words formed by using all the letters of the word 'AGAIN' such that all three vowels are taken in such a way that no two vowels are together and only consonants can be repeated as many times as we need, then $\frac{\lambda }{210}$ is

Q.

Point $p$ divides the line segment joining $A\left(-1,3\right)$ and $B\left(9,8\right)$ in ratio $k:1$. If $p$ lies on the line $x-y+2=0$, then the value of $k$ is

Q. The vertices of a triangle are (2, 1), (5, 2) and (4, 4). The lengths of the perpendicular from these vertices on the opposite sides are

1. 
2. 
3. 
4. 

Q. If the point $(2,a)$ divides the line segment joining the points $(-4,3)$ and $(6,3),$ then

1. the ratio is $2:3$
2. the ratio is $3:2$
3. the value of $a$ is $3$
4. the value of $a$ is $2$

Q. The line joining $(5,0)$ to $(10\cos \theta ,10\sin \theta )$ is divided internally in the ratio $2:3$ at $P$, then the locus of $P$ is

1. $x^2+2xy+y^2-6x=0$
2. $x+y-3=0$
3. $(x-3{)}^2+y^2=16$
4. $x^2=y-3$

Q. Find the ratio in which (the plane) $2x+3y+5z=1$ divides the line joining the points $(1,0,-3)$ and $(1,-5,7)$.

1. $1:2$
2. $3:1$
3. None of these
4. $2:3$

Q. The vertices of a triangle are $A(0,-6),$ $B(-6,0)$ and $C(1,1)$ respectively. Then coordinates of the excentre opposite to vertex $A$ are

1. $(-\frac{3}{2},-\frac{3}{2})$
2. $(-4,\frac{3}{2})$
3. $(\frac{3}{2},-\frac{3}{2})$
4. $(-4,6)$

Q. If $A=(1,2,3),B=(2,3,4)$ and $AB$ is produced upto $C$ such that $2AB=BC$, then $C=$

1. $(5,4,6)$
2. $(6,2,4)$
3. $(6,4,5)$
4. $(4,5,6)$

Q. If $P(x,y)$ divides the line segment joining $A(5,0)$ and $B(10\cos \theta ,10\sin \theta )$ internally in the ratio $2:3,$ then which of the following is (are) true?

1. Distance of $P$ from $(3,0)$ is always constant.
2. $x\in [0,3]$
3. $x^2+y^2-6x-7=0$
4. $y\in [-4,0]$

Q. The ratio in which line 3x - y - 9 divides the line segment joining point (1, 3) and (2, 7)

1. $-910$ externally
2. $910$ internally
3. $35$ externally
4. $35$ internally

Q. In what ratio the line $x+y=4$ divides the line joining the points $(-1,1)$ and $(5,7)?$

Q. If the $zx$-plane divides the line segment joining $(1,-1,5)$ and $(2,3,4)$ in the ratio $p:1$, then $p+1=$

1. $\frac{1}{3}$
2. $1$
3. $\frac{3}{4}$
4. $\frac{4}{3}$

Q. If point $(\frac{1}{2},y)$ lies on the line segment joining the points A (3 , -5) and B (-7, 9), then find the ratio in which P divides AB. also find the value of y .

Q. If the point P(2, 3) divides the line joining the points (5, 6) and (8, 9) then the ratio is

1. 1:2 internally
2. 1:2 externally
3. 2:1 internally
4. 2:1 externally

Q. The ratio which divides the line joining the points $(2,3)$ and $(4,2),$ also divides the segment joining the point $(1,2)$ and $(4,3)$ is

1. $1:2$
2. $(1:4)$
3. $(3:1)$
4. $1:1$

Q. $XOZ$ plane divides the join of $(2,3,1)$ and $(6,7,1)$ in the ratio

1. $3:7$
2. $2:7$
3. $-2:7$
4. $-3:7$